The global arithmetic intersection formula at trivial level

Let $f=\otimes_p f_p$ be a completely decomposed element of the indicated partial Hecke algebra at trivial level, and let $f'=\otimes_v f'_v$ be a Gaussian test function on $G'=\operatorname{Res}_{F/F_0}(\mathrm{GL}_{n-1}\times\mathrm{GL}_n)$ whose finite part is a smooth transfer of $f$. Let $\operatorname{Int}(f)$ be the normalized arithmetic intersection pairing and let $J$ and $\partial J$ denote the relevant global distribution and its derivative.

Global arithmetic intersection conjecture at trivial level. One has

$$\operatorname{Int}(f)=-\partial J(f')-J(f'_{\mathrm{corr}}),$$

where $f'_{\mathrm{corr}}\in C_c^\infty(G'({\mathbb A}_{F_0}))$ is a correction function; moreover, $f'$ may be chosen so that $f'_{\mathrm{corr}}=0$.

This is an arithmetic refinement of the global Gan–Gross–Prasad formula, relating an arithmetic intersection number to the derivative of an automorphic distribution. The source gives no evidence that it has been proved.